U.S. patent application number 13/538488 was filed with the patent office on 2014-01-02 for viscometer for newtonian and non-newtonian fluids.
This patent application is currently assigned to ROSEMOUNT INC.. The applicant listed for this patent is Roger Kenneth Pihlaja. Invention is credited to Roger Kenneth Pihlaja.
Application Number | 20140005957 13/538488 |
Document ID | / |
Family ID | 48940923 |
Filed Date | 2014-01-02 |
United States Patent
Application |
20140005957 |
Kind Code |
A1 |
Pihlaja; Roger Kenneth |
January 2, 2014 |
VISCOMETER FOR NEWTONIAN AND NON-NEWTONIAN FLUIDS
Abstract
A viscometer comprises a plurality of capillary tubes connected
in series with a mass flow meter. The capillary tubes are smooth,
straight, and unimpeded, and each has a different known, constant
diameter. Differential pressure transducers sense differential
pressure across measurement lengths of each capillary tube, and the
mass flow meter senses fluid mass flow rate and fluid density. A
data processor connected to the mass flow meter and the
differential pressure transducers computes viscosity parameters of
fluid flowing through the viscometer using non-Newtonian fluid
models, based on the known, constant diameters and measurement
lengths of each capillary tube, the sensed differential pressures
across each measurement length, the fluid mass flow rate, and the
fluid density.
Inventors: |
Pihlaja; Roger Kenneth;
(Spring Park, MN) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Pihlaja; Roger Kenneth |
Spring Park |
MN |
US |
|
|
Assignee: |
ROSEMOUNT INC.
Chanhassen
MN
|
Family ID: |
48940923 |
Appl. No.: |
13/538488 |
Filed: |
June 29, 2012 |
Current U.S.
Class: |
702/50 |
Current CPC
Class: |
G01F 1/84 20130101; G01N
2011/0026 20130101; G01N 11/08 20130101 |
Class at
Publication: |
702/50 |
International
Class: |
G01N 11/08 20060101
G01N011/08; G06F 19/00 20110101 G06F019/00 |
Claims
1. A viscometer comprising: a first capillary tube having a first
diameter D.sub.1 and a first tube length L.sub.Tot1; a first
differential pressure transducer operating across a first
measurement length L.sub.1 of the first capillary tube to sense a
first differential pressure .DELTA.P.sub.1, the first measurement
length L.sub.1 extending across a smooth, straight, and unimpeded
portion of the first capillary tube configured to produce steady
state laminar flow; a second capillary tube fluidly connected in
series after the first capillary tube and having a second diameter
D.sub.2.noteq.D.sub.1 and a second tube length L.sub.Tot2; a second
differential pressure transmitter operating across a second
measurement length L.sub.2 of the second capillary tube to sense a
second differential pressure .DELTA.P.sub.2, the second measurement
length L.sub.2 extending across a smooth, straight, and unimpeded
portion of the second capillary tube configured to produce steady
state laminar flow; a mass flow meter fluidly connected in series
after the second capillary tube, and capable of sensing fluid
density .rho. and fluid mass flow rate m; and a processor in data
communication with the mass flow meter, and capable of computing
viscosity parameters of fluid flowing through the first capillary
tube, the second capillary tube, and the mass flow meter using
non-Newtonian fluid models, based on D.sub.1, D.sub.2, L.sub.1,
L.sub.2, .DELTA.P.sub.1, .DELTA.P.sub.2, .rho., and m.
2. The viscometer of claim 1, wherein the processor is also capable
of computing the Newtonian viscosity of fluid flowing through the
first capillary tube, the second capillary tube, the second
capillary tube, an the mass flow meter based on D.sub.1, D.sub.2,
L.sub.1, L.sub.2, .DELTA.P.sub.1, .DELTA.P.sub.2, .rho., and m.
3. The viscometer of claim 1, wherein the first tube length
L.sub.Tot1 is greater than or equal to L.sub.1+0.07D.sub.1
[Re].sub.1, and L.sub.Tot2 is greater than or equal to
L.sub.2+0.07D.sub.2 [Re].sub.2, where [Re].sub.1 is the Reynolds
number of fluid flowing through the first capillary tube and
[Re].sub.2 is the Reynolds number of fluid flowing through the
second capillary tube.
4. The viscometer of claim 1, wherein the processor is configured
to model the fluid as a Bingham plastic, and wherein the computed
viscosity parameters are an apparent viscosity .mu..sub.A and a
critical shear stress .tau..sub.0.
5. The viscometer of claim 1, wherein the processor is configured
to model the fluid as an Ostwald-de Waele fluid, and wherein the
computed viscosity parameters are an apparent viscosity .mu..sub.A
and an exponential degree of deviation from Newtonian behavior
n.
6. The viscometer of claim 1, further comprising: a third capillary
tube fluidly connected in series after the first and second
capillary tubes, and having a third diameter D.sub.3.noteq.D.sub.1
or D2 and a third tube length L.sub.Tot3; and a third differential
pressure transmitter operating across a third measurement length
L.sub.3 of the third capillary tube to sense a third differential
pressure .DELTA.P.sub.3, the third measurement length L.sub.3
extending across a smooth, straight, and unimpeded portion of the
third capillary tube configured to produce steady state laminar
flow; and wherein the processor computes viscosity parameters based
on D.sub.3, L.sub.3 and .DELTA.P.sub.3, in addition to D.sub.1,
D.sub.2, L.sub.1, L.sub.2, .DELTA.P.sub.1, .DELTA.P.sub.2, .rho.,
and m.
7. The viscometer of claim 6, wherein the processor is configured
to model the fluid as an Ellis fluid with, and wherein the computed
viscosity parameters are .alpha., .phi..sub.0, and .phi..sub.1 of
the Ellis fluid equation .PHI. 0 .tau. rz + .PHI. 1 ( .tau. rz )
.alpha. = - V z r . ##EQU00019##
8. The viscometer of claim 6, wherein the processor is configured
to model the fluid as a Herschel-Bulkley fluid, and wherein the
computed viscosity parameters are a critical shear stress
.tau..sub.0, an apparent viscosity .mu..sub.A, and an exponential
degree of deviation from Newtonian behavior n.
9. The viscometer of claim 9, wherein the processor is configured
to compute .tau..sub.0, .mu..sub.A, and n by iterating alternately
between solving for .tau..sub.0 and .mu..sub.A using a Bingham
plastic model, and solving for .mu..sub.A and n using a Ostwald-de
Waele model.
10. The viscometer of claim 1, further comprising a temperature
sensor which produces a sensed fluid temperature used by the
processor to compute the viscosity parameters.
11. The viscometer of claim 1, wherein the mass flow meter is a
Coriolis effect mass flow meter.
12. The viscometer of claim 11, wherein one of the first capillary
tube and the second capillary tube is incorporated into the
Coriolis effect mass flow meter.
13. A method for characterizing viscosity of a fluid, the method
comprising: sensing a first differential pressure of the fluid
across a first length of a smooth, straight, and unimpeded first
capillary having a constant first diameter; sensing a second
differential pressure of the fluid across a second length of a
smooth, straight, and unimpeded second capillary fluidly connected
in series with the first capillary, and having a constant second
diameter; sensing fluid density and fluid mass flow rate at a mass
flow meter fluidly connected in series with the second capillary;
computing adjustable viscosity parameters of a non-Newtonian fluid
model using the first and second capillary lengths, the first and
second diameters, the sensed first and second differential
pressures, the fluid density, and the fluid mass flow rate; and
outputting the computed adjustable viscosity parameters in an
output signal.
14. The method of claim 13, wherein the non-Newtonian fluid model
is a Bingham plastic model, and wherein solving for adjustable
viscosity parameters comprises solving for apparent viscosity
.mu..sub.A and a critical shear stress .tau..sub.0.
15. The method of claim 13, wherein the non-Newtonian fluid model
is an Ostwald-de Waele model, and wherein solving for adjustable
viscosity parameters comprises solving for apparent viscosity
.mu..sub.A and an exponential degree of deviation from Newtonian
behavior n.
16. The method of claim 13, wherein the non-Newtonian fluid model
is an Ellis model, and wherein solving for adjustable viscosity
parameters comprises solving for .tau., .tau..sub.0, and
.phi..sub.1 of the Ellis fluid equation .PHI. 0 .tau. rz + .PHI. 1
( .tau. rz ) .alpha. = - V z r . ##EQU00020##
17. The method of claim 13, wherein the non-Newtonian fluid model
is a Herschel-Bulkley model, and wherein solving for adjustable
viscosity parameters comprises solving for critical shear stress
.tau..sub.0, an apparent viscosity .mu..sub.A, and an exponential
degree of deviation from Newtonian behavior n.
18. The viscometer of claim 17, wherein solving for .tau..sub.0,
.mu..sub.A, and comprises iterating alternately between solving for
.tau..sub.0 and .mu..sub.A using a Bingham plastic model, and
solving for .mu..sub.A and n using a Ostwald-de Waele model.
19. A viscometer comprising: first capillary tube coupled to a
first differential pressure sensor configured to sense a first
differential pressure across a steady state region of the first
capillary tube; a second capillary tube fluidly connected in series
with the first capillary tube, and coupled to a second differential
pressure sensor configured to sense a second differential pressure
across a steady state region of the second capillary tube; a sensor
device fluidly connected in series with the first and second
capillary tube, and capable of sensing fluid mass flow rate and
fluid density; and a data processor which computes a plurality of
viscosity parameters of fluid passing through the first capillary
tube, the second capillary tube, and the sensor device based on the
mass flow rate, the density, the differential pressure across each
capillary tube, and the dimensions of each capillary tubes.
20. The viscometer of claim 19, wherein the plurality of viscosity
parameters are free parameters of a non-Newtonian fluid model
selected from the group comprising Bingham plastic, Ostwald-de
Waele, an Ellis, or a Herschel-Bulkley fluid models.
21. The viscometer of claim 20, further comprising a memory
configured to store: a plurality of algorithms for computing the
viscosity parameters using any of a plurality of the group of fluid
models; and a fluid model selection designating one of the
plurality of algorithms to be used to compute the viscosity
parameters.
22. The viscometer of claim 21, wherein the data processor is a
part of a process transmitter configured to report the viscosity
parameters to a central controller.
23. The viscometer of claim 22, wherein the viscometer is
configured to fit in-line into an industrial process flow.
Description
BACKGROUND
[0001] The present invention relates generally to viscosity
measurement, and more particularly to a viscometer capable of
handling both Newtonian and non-Newtonian fluids.
[0002] Fluid viscosity is a critical and commonly measured
parameter in many industrial processes. A variety of viscometer
designs are used in such processes, typically by diverting a small
quantity of process fluid from a primary process flow path through
a viscometer connected in parallel with the primary process flow
path. A few in-line designs instead allow viscometers to be located
directly in the primary flow path, obviating the need to divert
process fluid. Most conventional industrial viscometers utilize
rotating parts in contact with process fluids, and consequently
require bearings and seals to prevent fluid from leaking. In
applications involving harsh, corrosive or abrasive fluids, such
viscometers may require frequent maintenance.
[0003] Conventional industrial process viscometers are well-suited
to measuring Newtonian fluids (wherein viscosity is constant). A
wide range of industrial applications, however, handle slurries,
pastes, and plastics which behave in a non-Newtonian fashion, and
which conventional viscometers are not equipped to measure. Such
industrial applications include oil field drilling (e.g. handling
drilling mud), paste or plastic manufacture (e.g. handling
cosmetics or polymers, or building products such as paint, plaster,
or mortar), refining (e.g. handling lube or fuel oil), and food
processing.
[0004] The viscosity of Newtonian fluids in Couette flow (i.e. flow
between two parallel plates, one of which is moving relative to the
other) is described by:
F A = .tau. = - .mu. u y [ Equation 1 ] ##EQU00001##
where F is shear force, A is the cross-sectional area of each
plane, .tau. is shear stress (or equivalently momentum flux), .mu.
is viscosity, and du/dy is shear rate. Extrapolating from this
formula yields the following relation between shear stress, shear
rate, and viscosity within a tube carrying Newtonian fluid
flow:
.tau. rz = - .mu. V z r Newtonian Fluid [ Equation 2 ]
##EQU00002##
where .tau..sub.rz is shear stress in the radial (r) direction,
normal to the axis of the tube (i.e. the z direction), and
dV.sub.z/dr is shear rate in the z direction with respect to r.
[0005] Equation 2 describes Newtonian fluids (and fluids in
substantially Newtonian regimes), wherein viscosity (.mu.) does not
vary as a function of shear rate. Non-Newtonian fluids, however,
may become more viscous ("shear thickening" or "dilatant" fluids)
or less viscous ("shear thinning" or "pseudoplastic" fluids") as
shear rate increases. A variety of empirical models have been
developed to describe non-Newtonian fluid behavior, including the
Bingham plastic, Ostwald-de Waele, Ellis, and Herschel-Bulkley
models (described in greater depth below). FIG. 1 provides an
illustration of shear stress as a function of shear rate for each
of these models. For the most part these models have no theoretical
basis, but each has been shown to be accurate describe a subset of
non-Newtonian fluids.
[0006] The Bingham plastic model utilizes two viscosity-related
parameters, "shear stress" and "apparent viscosity," rather than a
single Newtonian viscosity parameter. Bingham plastics do not flow
unless subjected to sufficient shear stress. Once a critical shear
stress .tau..sub.0 is exceeded, Bingham plastics behave in a
substantially Newtonian fashion, exhibiting a constant apparent
viscosity .mu..sub.A, as follows:
.tau. rz = .tau. 0 - .mu. A V z r Bingham Plastic [ Equation 3 ]
##EQU00003##
[0007] Like the Bingham plastic model, the Ostwald-de Waele model
provides a two-parameter description of fluid viscosity. The
Ostwald-de Waele model is suited to "power law" fluids wherein
shear stress is a power (rather than a linear) function of shear
rate. Ostwald-de Waele fluids behave as follows:
.tau. rz = .mu. A [ - V z r ] n Ostwald - de Waele Fluid [ Equation
4 ] ##EQU00004##
where .mu..sub.A is apparent viscosity, and n is a degree of
deviation from Newtonian fluid behavior, with n<1 corresponding
to a pseudoplastic fluid, and n>1 corresponding to a dilatant
fluid.
[0008] The Ellis model uses three, rather than two, adjustable
parameters to characterize fluid viscosity. The Ellis model
describes shear rate as a function of shear stress, as follows:
.PHI. 0 .tau. rz + .PHI. 1 ( .tau. rz ) .alpha. = - V z r Ellis
Fluid [ Equation 5 ] ##EQU00005##
where .alpha., .phi..sub.0, and .phi..sub.1 are adjustable
parameters. The Ellis model combines power law and linear
components scaled by constants .phi..sub.0, and .phi..sub.1, with
.alpha.>1 corresponding to a pseudoplastic fluid and
.alpha.<1 corresponding to a dilatant fluid.
[0009] The Herschel-Bulkley fluid model combines the power law
behavior of Ostwald-de Waele fluids with the rigidity of Bingham
plastics below a critical shear stress, and uses three adjustable
parameters. The Herschel-Bulkley model is particularly well suited
to describing the slurries and muds handled in oil and gas drilling
applications. According to the Herschel-Bulkley model,
.tau. rz = .tau. 0 - .mu. A [ V z r ] n Herschel - Bulkley Fluid [
Equation 6 ] ##EQU00006##
where .tau..sub.0 is critical shear stress, .mu..sub.A is apparent
viscosity, and n is a degree of deviation from Newtonian fluid
behavior as described above with respect to the Ostwald-de Waele
fluid model (Equation 4).
[0010] Each of the models introduced above describes a class of
non-Newtonian fluids which are not well handled by conventional
industrial viscometers.
SUMMARY
[0011] The present invention is directed toward a viscometer
comprising a plurality of capillary tubes connected in series with
a mass flow meter. The capillary tubes are smooth, straight, and
unimpeded, and each has a different known, constant diameter.
Differential pressure transducers sense differential pressure
across measurement lengths of each capillary tube, and the mass
flow meter senses fluid mass flow rate and fluid density. A data
processor connected to the mass flow meter and the differential
pressure transducers computes viscosity parameters of fluid flowing
through the viscometer using non-Newtonian fluid models, based on
the known, constant diameters and measurement lengths of each
capillary tube, the sensed differential pressures across each
measurement length, the fluid mass flow rate, and the fluid
density. The present invention is further directed towards a method
for determining these viscosity parameters using the aforementioned
viscometer.
BRIEF DESCRIPTION OF THE DRAWINGS
[0012] FIG. 1 is a graph illustrating shear stress as a function of
shear rate according to several Newtonian and non-Newtonian fluid
models.
[0013] FIG. 2 is a schematic depiction of the viscometer of the
present invention.
[0014] FIG. 3 is a flow chart of a method for computing fluid
viscosity parameters of to the Herschel-Bulkley model.
DETAILED DESCRIPTION
[0015] In general, the present invention relates to an in-line
viscometer capable of handling any of a plurality of kinds of
Newtonian or non-Newtonian fluids, including Bingham plastics and
Ostwald-de Waele, Ellis, and Herschel-Bulkley fluids.
[0016] Viscometer Hardware
[0017] FIG. 2 depicts one illustrated embodiment of viscometer 10,
comprising process flow inlet 12, first capillary tube 14, joint
seals 16, connecting tubes 18, second capillary tube 20, third
capillary tube 22, Coriolis mass flow meter 24, process flow outlet
26, first differential pressure transducer 28, second differential
pressure transducer 30, third differential pressure transducer 32,
first isolation diaphragms 34a and 34b, second isolation diaphragms
36a and 36b, third isolation diaphragms 38a and 38b, and process
transmitter 40. Process transmitter 40 further comprises signal
processor 42, memory 44, data processor 46, and input/output block
48.
[0018] Pursuant to the embodiment of FIG. 2, First, second, and
third capillary tubes 14, 20, and 22 are smooth capillaries or
tubes that allow fluid flow to equilibrate into a steady state
shear distribution which does not vary as a function of axial
position along measurement lengths L.sub.1, L.sub.2, and L.sub.3.
Measurement lengths L.sub.1, L.sub.2, and L.sub.3 extend between
isolation diaphragms 34a and 34b, seals 36a and 36b, and 38a and
38b, respectively. Measurement lengths L.sub.1, L.sub.2, and
L.sub.3 are located in substantially the mid portions of capillary
tubes 14, 20, and 22. Each capillary tube 14, 20, and 22 has a
different known diameter D.sub.1, D.sub.2, and D.sub.3,
respectively. Capillary tubes 14, 20, and 22 are connected in
series with Coriolis mass flow meter 24, a conventional Coriolis
effect device which measures fluid mass flow rate m, fluid density
.rho., and fluid temperature T. Fluid enters first capillary tube
14 through process flow inlet 12, flows in series through second
capillary tube 20, third capillary tube 22, and Coriolis mass flow
meter 24, then exits viscometer 10 through process flow outlet 26.
Process flow inlet 12 and process flow outlet 26 are connecting
tubes or pipes which carry fluid from an industrial process, such
as fluid polymer from a polymerization process or waste slurry from
a drilling process. Viscometer 10 provides an in-line measurement
of viscosity, rather than measuring the viscosity of a diverted
fluid stream. This viscosity measurement takes the form of an
output signal S.sub.out containing a number of viscosity parameters
dependant on the fluid model used.
[0019] Although this Specification describes viscometer 10 as
having three capillary tubes (14, 20, and 22), a person skilled in
the art will recognize that additional capillary tubes may be
needed to compute all viscosity parameters for fluid models with a
large number of adjustable parameters. Similarly, fluid models with
fewer adjustable parameters (such as the Bingham plastic and
Ostwald-de Waele models, which have only two adjustable parameters,
or the Newtonian fluid model, which has only one) may require fewer
capillary tubes. Three capillary tubes are sufficient to compute
all viscosity parameters for the fluid models considered herein.
Although FIG. 2 depicts three capillary tubes, some embodiments of
the present invention may use two capillary tubes, or four or more
capillary tubes.
[0020] Pursuant to the embodiment of FIG. 2, connecting tubes 18
are pipes or tubes which join first capillary tube 14 to second
capillary tube 20, and second capillary tube 20 to third capillary
tube 22. Viscometer 10 is not sensitive to the shape or dimensions
of connecting tubes 18, and some embodiments of viscometer 10 may
lack one or more of the depicted connecting tubes, or include
additional connecting tubes not shown in FIG. 2. In some
embodiments, for instance, second capillary tube 20 may be
connected directly (i.e. without any connecting tube 18) to first
capillary tube 20 and/or third capillary tube 22. In other
embodiments, additional connecting tubes may be interposed between
process flow inlet 12 and first capillary tube 14, between third
capillary tube 22 and Coriolis mass flow meter 24, and/or between
Coriolis mass flow meter 24 and process flow outlet 26. Capillary
tubes 14, 20, and 22 are formed of a rigid material such as copper,
steel, or aluminum. The material selected for capillary tubes 14,
20, and 22 may depend on the process fluid, which in some
applications can be caustic, abrasive, or otherwise damaging to
some materials. Connecting tubes 18 may be formed of the same
material as capillary tubes 14, 20, and 22, or may be formed of a
less rigid material which is likewise resilient to the process
fluid.
[0021] First, second, and third differential pressure transducers
28, 30, and 32 are conventional differential pressure devices such
as capacitative differential pressure cells. Differential pressure
transducers 28, 30, and 32 measure differential pressure across
measurement lengths L.sub.1, L.sub.2, and L.sub.3 of capillary
tubes 14, 20, and 22, using isolation diaphragms 34, 36, and 38,
respectively. Isolation diaphragms 34, 36, and 38 are diaphragms
which transmit pressure from process fluid flowing through
capillary tubes 14, 20, and 22, to differential pressure
transducers 28, 30, and 32 via pressure lines such as closed oil
capillaries. Isolation diaphragms 34a and 34b are positioned at
opposite ends of measurement length L.sub.1, isolation diaphragms
36a and 36b are positioned at opposite ends of measurement length
L.sub.2, and isolation diaphragms 34a and 34b are positioned at
opposite ends of length L.sub.3. Differential pressure transducers
28, 30, and 32 produce differential pressure signals
.DELTA.P.sub.1, .DELTA.P.sub.2, and .DELTA.P.sub.3, which reflect
pressure change across measurement lengths L.sub.1, L.sub.2, and
L.sub.3, respectively.
[0022] Although the present Specification describes sensing
differential pressure directly via differential pressure cells, a
person skilled in the art will understand that differential
pressure could equivalently be measured in a variety of ways,
including using two or more absolute pressure sensors positioned
along each of measurement lengths L.sub.1, L.sub.2, and L.sub.3 of
capillary tubes 14, 20, and 22. The particular method of
differential pressure sensing selected may depend on the specific
application, and on process flow pressures.
[0023] In one embodiment, process transmitter 40 is an electronic
device which receives sensor signals from Coriolis mass flow meter
24 and differential pressure transducers 28, 30, and 32, receives
command signals from a remote monitoring/control room or center
(not shown), computes process fluid viscosity based on one or more
fluid models, and transmits this computed viscosity to the remote
monitoring/control room. Process transmitter 40 includes signal
processor 42, memory 44, data processor 46, and input/output block
48. Signal processor 44 is a conventional signal processor which
collects and processes sensor signals from differential Coriolis
mass flow meter 24 and pressure transducers 28, 30, and 32. Memory
44 is a conventional data storage medium such as a semiconductor
memory chip. Data processor 46 is a logic-capable device such as a
microprocessor. Input/output block 48 is a wired or wireless
interface which transmits, receives, and converts analog or digital
signals between process transmitter 40 and the remote
monitoring/control room.
[0024] Signal processor 42 collects and digitizes differential
pressure signals .DELTA.P.sub.1, .DELTA.P.sub.2, and .DELTA.P.sub.3
from differential pressure transducers 28, 30, and 32, and fluid
mass flow rate m, fluid density .rho., and fluid temperature T from
Coriolis mass flow meter 24. Signal processor 42 also normalizes
and adjusts these values as necessary to calibrate each sensor.
Signal processor 42 may receive calibration information or
instructions from data processor 46 or input/output block 48 (via
data processor 46).
[0025] Memory 44 is a conventional non-volatile data storage medium
which is loaded with measurement lengths L.sub.1, L.sub.2, and
L.sub.3 and diameters D.sub.1, D.sub.2, and D.sub.3. Memory 44
supplies these values to data processor 46 as needed. Memory 44 may
also store temporary data during viscosity computation, and
permanent or semi-permanent history data reflecting past viscosity
information, configuration information, or the like. In some
embodiments, memory 44 may be loaded with a plurality of algorithms
for computing viscosity of fluids according to multiple models
(e.g. Newtonian, Bingham plastic, Ostwald-de Waele, Ellis, or
Herschel-Bulkley). In such embodiments, memory 44 may further store
a model selection designating one of these algorithms for use at
the present time. This model selection can be provided by a user or
remote controller via input/output block 48, or may be made by data
processor 46. Some embodiments of process transmitter 40 may only
be configured to handle a single fluid model.
[0026] Data processor 46 computes one or more adjustable viscosity
parameters according to at least one fluid model introduced above,
using measurement lengths L.sub.1, L.sub.2, and L.sub.3 and
diameters D.sub.1, D.sub.2, and D.sub.3 from memory 44, and
differential pressures .DELTA.P.sub.1, .DELTA.P.sub.2, and
.DELTA.P.sub.3, fluid mass flow rate m, fluid density .rho., and
fluid temperature T from signal processor 42. The particular
adjustable viscosity parameters computed depend on the fluid model
selected, as discussed in greater detail below with respect to each
model. Using the Bingham plastic model, for instance, data
processor 46 would compute shear stress .tau..sub.0 and apparent
viscosity .mu..sub.A. As noted above, models with only two
adjustable parameters (e.g. the Bingham plastic and Ostwald-de
Waele models) will require data for only two of the three capillary
tubes provided. In such cases, L.sub.3, D.sub.3, and
.DELTA.P.sub.3, for instance, may be disregarded. Data processor 46
assembles all computed viscosity parameters into an output signal
S.sub.out, which input/output block 48 transmits to the remote
controller.
[0027] Input/output block 48 transmits output signal S.sub.out to
the remote controller, and receives commands from the remote
controller and any other external sources. Where data processor 46
provides output signal S.sub.out in a format not appropriate for
transmission, input/output block 48 may also convert S.sub.out into
an acceptable analog or digital format. Some embodiments of
input-output block 48 communicate with the remote controller via a
wireless transceiver, while others may use wired connections.
[0028] Data processor 46 computes viscosity parameters for a
selected fluid model using variations on the Hagan-Poiseuille
equation. For Newtonian fluids, the Hagan-Poiseuille equation
states that:
m .rho. = .pi. ( .DELTA. P ) ( D / 2 ) 4 8 .mu. L Newtonian Hagan -
Poiseuille [ Equation 7 ] ##EQU00007##
where m is fluid mass flow rate, .rho. is fluid density, .mu. is
viscosity, and .DELTA.P is a pressure differential across a single
capillary of length L and diameter D. By measuring differential
pressure across measurement lengths L.sub.1, L.sub.2, and L.sub.3
of first, second, and third capillary tubes 14, 20, 22 (each of
which has a different known diameter D), viscometer 10 is able
solve non-Newtonian variants of the Hagan-Poiseuille equation with
multiple viscosity parameters, as described in greater detail
below.
[0029] The Hagan-Poiseuille equation assumes fully developed,
steady-state, laminar flow through a round cross-section
constant-diameter capillary tube with no slip between fluid and the
capillary wall. To ensure that all of these assumptions hold true,
capillary tubes 14, 20, and 22 must be entirely straight, smooth,
and devoid of any features which might disrupt steady-state flow.
In addition, capillary tubes 14, 20, and 22 must be long enough
that changes in tube geometry near the ends of capillary tubes 14,
20, and 22 (e.g. turns in connecting tubes 18, or changes in tube
diameter) have negligible effect on the behavior of fluid within
passing through measurement lengths L.sub.1, L.sub.2, and L.sub.3
of these capillary tubes. Accordingly, each capillary tube extends
a buffer length L.sub.E to either end of each measurement length,
to minimize the effect of such changes in geometry. This buffer
length L.sub.E is:
L.sub.E.gtoreq.0.035*D*[Re] Buffer Length [Equation 7]
where D is the diameter of the appropriate capillary tube, and [Re]
is the Reynolds number of the process fluid within the capillary
tube. [Re] is a dimensionless quantity which provides a measure of
turbulence within the flowing process fluid. [Re] can be calculated
for each fluid model as known in the art, but is in any case less
than 2100 for laminar flow. Generally, each capillary tube 14, 20,
and 22 has a total length L.sub.Tot greater than or equal to
L+2L.sub.E, i.e.
L.sub.Tot1.gtoreq.L.sub.1+2L.sub.E1=L.sub.1+0.07D.sub.1[Re].sub.1,
L.sub.Tot2.gtoreq.L.sub.2+2L.sub.2=+0.07D.sub.2[Re].sub.2, etc.
[0030] Bingham plastics and Herschel-Bulkley fluids will not flow
if shear stress does not exceed a critical shear stress
.tau..sub.0. To carry such fluids, capillary tubes 14, 20, and 22
must be constructed such that
.tau. 0 < D * .DELTA. P total 4 L total [ Equation 8 ]
##EQU00008##
where D is the diameter of the capillary tube, L.sub.total is the
total length of the capillary tube, and .DELTA.P.sub.total is the
total pressure drop across the capillary tube.
[0031] Fluid Model Solutions
[0032] As noted above, in some embodiments memory 44 may store
algorithms for solving for parameters of various fluid models,
based on measurement lengths L.sub.1, L.sub.2, and L.sub.3,
diameters D.sub.1, D.sub.2, and D.sub.3, differential pressures
.DELTA.P.sub.1, .DELTA.P.sub.2, and .DELTA.P.sub.3, fluid mass flow
rate m, fluid density .rho., and fluid temperature T.
Alternatively, data processor 46 may be hardwired to solve for
parameters of one or more fluid models. These parameters are then
transmitted to the remote monitoring/control room as a part of
output signal S.sub.out, and may be stored locally or provided to
other devices or users in some embodiments. Although particular
parameters, and the algorithms used to solve for them, vary from
model to model, all parameters of all models considered herein can
be computed using no more than three capillary tubes (i.e.
capillary tubes 14, 20, and 22) of known diameter and measurement
length. A person skilled in the art will understand that, although
the Newtonian, Bingham plastic, Ostwald-de Waele, Ellis, and
Herschel-Bulkley models are discussed in detail herein, other fluid
models might additionally or alternatively be utilized, with
viscometer 10 incorporating additional capillary tubes as needed
for models having a larger number of free parameters.
[0033] For Bingham plastics, the Hagan-Poiseuille equation
becomes:
m .rho. = .pi..DELTA. P [ D / 2 ] 4 8 .mu. A L ( 1 - 4 3 ( .tau. 0
/ .tau. R ) + 1 3 ( .tau. 0 / .tau. R ) 4 ) ; where .tau. R =
.DELTA. P [ D / 2 ] 2 L , i . e . .tau. 1 = .DELTA. P 1 [ D 1 / 2 ]
2 L 1 , .tau. 2 = .DELTA. P 2 [ D 2 / 2 ] 2 L 2 , etc . Bingham
Plastic Hagan Poiseuille [ Equations 9 ] ##EQU00009##
for the domain within which the Bingham plastic model is continuous
(i.e. for .tau..sub.R>.tau..sub.0, under which conditions
Bingham plastics flow). As stated previously, m is fluid mass flow
rate, .rho. is fluid density, D is capillary tube diameter, L is
measurement length, .tau..sub.0 is the critical shear stress
required for fluidity, and .mu..sub.A is the apparent viscosity of
the Bingham plastic for .tau.>.tau..sub.0.
[0034] .DELTA.P is a linear function of .tau..sub.R, such that:
.DELTA. P = C 1 .tau. R + .DELTA. vP 0 ; where [ Equation 10 ] C 1
= .DELTA. P 2 - .DELTA. P 1 .tau. 2 - .tau. 1 and [ Equation 11 ]
.DELTA. P 0 = .DELTA. P 1 - C 1 .tau. 1 [ Equation 12 ]
##EQU00010##
Accordingly, it is possible to solve for the two viscosity
parameters of the Bingham plastic model--critical shear stress
.tau..sub.0 and apparent viscosity .mu..sub.A--by substituting into
Equations 9, which yields:
.tau. 0 = .DELTA. P 0 [ D 1 / 2 ] 2 L 1 = D 1 4 L 1 ( .DELTA. P 1 -
C 1 .tau. 1 ) ; and [ Equation 13 ] .mu. A = .DELTA. P 1 [ D 1 / 2
] 4 .pi..rho. 8 mL 1 ( 1 - 4 3 ( .tau. 0 / .tau. 1 ) + 1 3 ( .tau.
0 / .tau. 1 ) 4 ) [ Equation 14 ] ##EQU00011##
[0035] When the model selection stored in memory 44 designates the
Bingham plastic model (or in embodiments wherein data processor 46
is hardcoded for Bingham plastics), data processor 46 computes
.tau..sub.0 and .mu..sub.A using this solution.
[0036] For Ostwald-de Waele fluids, the Hagan-Poiseuille equation
becomes:
m .rho. = .pi. [ D / 2 ] 3 3 n + 1 ( [ D / 2 ] .DELTA. P 2 .mu. A L
) 1 n Ostwald - de Waele Hagan - Poiseuille [ Equation 15 ]
##EQU00012##
where m is fluid mass flow rate, .rho. is fluid density, D is
capillary tube diameter, L is measurement length, .mu..sub.A is
apparent viscosity, and n is a degree of deviation from Newtonian
fluid behavior, as described previously. Substituting measurement
lengths L, differential pressures .DELTA.P, and capillary tube
diameters D for two capillary tubes (which may be any of capillary
tubes 14, 20, or 22) yields two equations:
m .rho. = .pi. [ D 1 / 2 ] 3 3 n + 1 ( [ D 1 / 2 ] .DELTA. P 1 2
.mu. A L 1 ) 1 n m .rho. = .pi. [ D 2 / 2 ] 3 3 n + 1 ( [ D 2 / 2 ]
.DELTA. P 2 2 .mu. A L 2 ) 1 n } [ Equations 16 ] ##EQU00013##
which can be solved simultaneously for n and .mu..sub.A,
yielding:
n = ln ( D 1 .DELTA. P 1 L 2 ) - ln ( D 2 .DELTA. p 2 L 1 ) 3 ln (
D 2 / D 1 ) [ Equation 17 ] .mu. A = [ D 1 / 2 ] .DELTA. P 1 2 L 1
( m ( 3 n + 1 ) / .pi..rho. [ D 1 / 2 ] 3 ) n [ Equation 18 ]
##EQU00014##
[0037] When the model selection stored in memory 44 designates the
Ostwald-de Waele model (or in embodiments wherein data processor 46
is hardcoded for the Ostwald-de Waele model), data processor 46
computes n and .mu..sub.A using this solution. The Ostwald-de Waele
model and the Bingham plastic model have only two free parameters,
and thus require only two capillary tubes for a complete solution.
Consequently, embodiments of viscometer 10 intended only to utilize
these and other two-dimensional models could dispense with third
capillary tube 22. Alternatively, viscometer 10 separately compute
fluid parameters using more than one combination of capillary tubes
(e.g. capillary tubes 14 and 20, capillary tubes 14 and 22, and
capillary tubes 20 and 22), and compare the results of these
computations--which should be substantially identical--to verify
that viscometer 10 is correctly calibrated and functioning.
[0038] The Ellis and Herschel-Bulkley models utilize three
viscosity parameters. Consequently, all three capillary tubes 14,
20 and 22 of the embodiment depicted in FIG. 2 are needed to solve
for these parameters, and more than three capillary tubes would be
required to produce redundant solutions for verification. For Ellis
fluids, the Hagan-Poiseuille equation becomes:
m .rho. = .pi..PHI. 0 .DELTA. P [ D / 2 ] 4 8 L + .pi..PHI. 1 [ D /
2 ] 3 .alpha. + 3 ( [ D / 2 ] .DELTA. P 2 L ) .alpha. Ellis Hagan -
Poiseuille [ Equation 19 ] ##EQU00015##
where m is fluid mass flow rate, .rho. is fluid density, D is
capillary tube diameter, L is measurement length, and .alpha.,
.phi..sub.0, and .phi..sub.1 are adjustable parameters of the Ellis
model as described previously. Substituting measurement lengths L,
differential pressures .DELTA.P, and capillary tube diameters D for
each capillary tubes 14, 20 and 22 yields three equations:
m .rho. = .pi..PHI. 0 .DELTA. P 1 [ D 1 / 2 ] 4 8 L 1 + .pi..PHI. 1
[ D 1 / 2 ] 3 .alpha. + 3 ( [ D 1 / 2 ] .DELTA. P 1 2 L 2 ) .alpha.
m .rho. = .pi..PHI. 0 .DELTA. P 2 [ D 2 / 2 ] 4 8 L 2 + .pi..PHI. 1
[ D 2 / 2 ] 3 .alpha. + 3 ( [ D 2 / 2 ] .DELTA. P 2 2 L 2 ) .alpha.
m .rho. = .pi..PHI. 0 .DELTA. P 3 [ D 3 / 2 ] 4 8 L 3 + .pi..PHI. 1
[ D 3 / 2 ] 3 .alpha. + 3 ( [ D 3 / 2 ] .DELTA. P 3 2 L 3 ) .alpha.
} [ Equations 20 ] ##EQU00016##
[0039] There is no closed-form analytic solution for the system of
equations 20. If the model selection stored in memory 44 designates
the Ellis model (or if data processor 46 is hardcoded for the Ellis
model), data processor 46 can simultaneously solve equations 20 for
.alpha., .phi..sub.0, and .phi..sub.1 using any of a plurality of
conventional iterative computational techniques.
[0040] For Herschel-Bulkley fluids, the Hagan-Poiseuille equation
becomes:
.DELTA. P L = 4 .mu. A D [ 32 m .rho..pi. D 3 ] n [ 3 n + 1 4 n ] n
[ 1 1 - X ] [ 1 1 - aX - bX 2 - cX ] n ; with a = 1 2 n + 1 ; b = 2
n ( n + 1 ) ( 2 n + 1 ) ; c = 2 n 2 ( n + 1 ) ( 2 n + 1 ) ; and X =
4 L .tau. 0 D .DELTA. P ; i . e . X 1 = 4 L 1 .tau. 0 D 1 .DELTA. P
1 ; X 2 = 4 L 2 .tau. 0 D 2 .DELTA. P 2 ; etc . Herschel - Bulkley
Hagan - Poiseuille [ Equations 21 ] ##EQU00017##
where m is fluid mass flow rate, .rho. is fluid density, D is
capillary tube diameter, L is measurement length, .tau..sub.0 is
critical shear stress, .mu..sub.A is apparent viscosity, and n is a
degree of deviation from Newtonian fluid behavior. Substituting
measurement lengths L, differential pressures .DELTA.P, and
capillary tube diameters D for each capillary tubes 14, 20 and 22
yields three equations:
[ Equations 22 ] .DELTA. P 1 L 1 = 4 .mu. 0 D 1 [ 32 m .rho..pi. D
1 3 ] n [ 3 n + 1 4 n ] n [ 1 1 - X 1 ] [ 1 1 - aX 1 - bX 1 2 - cX
1 3 ] n .DELTA. P 2 L 2 = 4 .mu. 0 D 2 [ 32 m .rho..pi. D 2 3 ] n [
3 n + 1 4 n ] n [ 1 1 - X 2 ] [ 1 1 - aX 2 - bX 2 2 - cX 2 3 ] n
.DELTA. P 3 L 3 = 4 .mu. 0 D 3 [ 32 m .rho..pi. D 3 3 ] n [ 3 n + 1
4 n ] n [ 1 1 - X 3 ] [ 1 1 - aX 3 - bX 3 2 - cX 3 3 ] n }
##EQU00018##
[0041] As with the Ellis model, there is no closed-form analytic
solution for the system of equations 22. If the model selection
stored in memory 44 designates the Herschel-Bulkley model (or if
data processor 46 is hardcoded for the Herschel-Bulkley model),
data processor 46 solves equations 22 for .tau..sub.0, .mu..sub.A,
and n computationally. Because the Herschel-Bulkley model combines
the power law behavior of Ostwald-de Waele fluids with the critical
shear stress discontinuity of Bingham plastics, a particularly
efficient computational simultaneous solution of equations 22 uses
the previously discussed analytic solutions to the Ellis and
Bingham plastic models to iteratively improve upon estimates of
.tau..sub.0, .mu..sub.A, and n.
[0042] FIG. 3 is a flow chart of method 100, which provides an
iterative computational solution to Equations 22. First, data
processor 46 retrieves measurement lengths L.sub.1, L.sub.2, and
L.sub.3, and capillary tube diameters D.sub.1, D.sub.2, D.sub.3
from memory 44, and differential pressures .DELTA.P.sub.1,
.DELTA.P.sub.2, and .DELTA.P.sub.3, fluid mass flow rate m, and
fluid density .rho. from Coriolis mass flow meter 24. (Step S1).
Next, data processor 46 approximates process fluid flow as a
Bingham plastic and solves for initial values of .DELTA.P.sub.0,
.mu..sub.A, and .tau..sub.0 using Equations 12 and 13,
respectively. (Step S2). Data processor 46 then produces adjusted
differential pressures
.DELTA.P.sub.1A=.DELTA.P.sub.1-.DELTA.P.sub.0,
.DELTA.P.sub.2A=.DELTA.P.sub.2-.DELTA.P.sub.0 and
.DELTA.P.sub.3A=.DELTA.P.sub.3-.DELTA.P.sub.0. (Step S3).
Substituting adjusted differential pressures .DELTA.P.sub.1A,
.DELTA.P.sub.2A, and .DELTA.P.sub.3A for measured differential
pressures .DELTA.P.sub.1, .DELTA.P.sub.2, and .DELTA.P.sub.3 allows
data processor 46 to approximate process fluid as an Ostwald-de
Waele fluid. Data processor 46 solves for n and .mu..sub.A with
Equations 17 and 18, respectively, with all possible combinations
of .DELTA.P.sub.1A, .DELTA.P.sub.2A, and .DELTA.P.sub.3A (i.e.
.DELTA.P.sub.1A and .DELTA.P.sub.2A, .DELTA.P.sub.1A and
.DELTA.P.sub.3A, and .DELTA.P.sub.2A and .DELTA.P.sub.3A), and
utilizes the mean of these solution values as n and .mu..sub.A.
(Step S4). Data processor 46 then calculates a next estimate of
.DELTA.P.sub.0 using these values of n and .mu..sub.A. (Step S5).
On the first iteration of method 100, (checked in Step S6) data
processor 46 then stores the current estimates of .tau..sub.0,
.mu..sub.A, and n in memory 44. (Step S7). On subsequent iterations
(checked in Step S6), data processor 46 compares the latest
estimates of .tau..sub.0, .mu..sub.A, and n to stored values to
determine whether .tau..sub.0, .mu..sub.A, and n have converged.
(Step S8). If the differences between stored values and the latest
estimates are negligible (or, more generally, if these differences
fall below a predefined threshold), data processor 46 passes the
latest values of .tau..sub.0, .mu..sub.A, and n to input/output
block 48, which transmits output signal S.sub.out, to the remote
controller and any other intended recipients. (Step S9). Otherwise,
processor 46 stores the latest estimates of .tau..sub.0,
.mu..sub.A, and n in memory 44 (Step S7), and computes new
estimates of .tau..sub.0 and .mu..sub.A using equations 12 and 13,
and the newly .DELTA.P.sub.0 estimate of Step S5. (Step S10). These
new estimates of .tau..sub.0 and .mu..sub.A are used to produce new
estimates of n and .mu..sub.A from Equations 17 and 18, as method
100 repeats itself.
[0043] By iteratively alternating between approximating a
Herschel-Bulkley fluid as a Bingham plastic and an Ostwald-de Waele
fluid, method 100 is able to rapidly converge upon a highly
accurate computational solution to Equations 22. A person skilled
in the art will understand, however, that other computational
methods could also be used to determine critical shear stress
.tau..sub.0, apparent viscosity .mu..sub.A, and degree of deviation
from Newtonian behavior n.
[0044] The viscosities of many fluids are temperature-dependant.
For industrial processes which operate at substantially constant
temperature, this temperature dependence may typically be ignored
Likewise, some applications may require that viscosity be measured
at a fixed temperature. To accomplish this, process fluid may be
pumped to a heat exchanger, or viscometer 10 maybe mounted in a
regulated constant temperature bath. Although the particular
details of viscosity temperature-dependence are not discussed
herein, data processor 46 may receive temperature readings from
within viscometer 10 for applications wherein considerable
temperature variation is expected. In particular, the present
Specification has described Coriolis mass flow meter 24 as
providing a measurement of fluid temperature T. A person having
ordinary skill in the art will recognize that temperature sensors
may alternatively or additionally be integrated into other
locations within viscometer 10.
[0045] As noted above, viscometer 10 may contain more or fewer
capillary tubes than the three (capillary tubes 14, 20, and 22)
described herein. In particular, embodiments of viscometer 10
suited for two-dimensional fluid models may feature only two
capillary tubes, while embodiments suited for four (or
more)--dimensional fluid models will require additional capillary
tubes. In addition, some embodiments of viscometer 10 may dispense
with one capillary tube by measuring a pressure drop across
Coriolis mass flow meter 24. Because Coriolis mass flow meter 24
does not provide the perfectly straight, smooth, and unimpeded
fluid path required to ensure steady-state laminar fluid flow, the
Hagan-Poiseuille equation would not accurately describe fluid
behavior through such a system, and computed viscosity parameter
accuracy would accordingly suffer. For may applications, however, a
slight decrease in accuracy may be an acceptable trade for making
viscometer 10 less expensive and more compact.
[0046] Viscometer 10 can be used to determine the viscosity of
Newtonian fluids, but more significantly allows viscosity
parameters to be measured with high accuracy for various
non-Newtonian fluid models, including but not limited to the
Bingham plastic, Ellis, Ostwald-de Waele, and Herschel-Bulkley
models. As described above, process transmitter 40 may be
manufactured with the capacity to handle multiple fluid models,
allowing viscometer 10 to be adapted to a range of fluid
applications by specifying a particular model, without replacing
any hardware. Viscometer 10 operates in-line with industrial
processes stream, and therefore need not divert process fluid away
from a process stream in order to produce an accurate measure of
process fluid viscosity.
[0047] While the invention has been described with reference to an
exemplary embodiment(s), it will be understood by those skilled in
the art that various changes may be made and equivalents may be
substituted for elements thereof without departing from the scope
of the invention. In addition, many modifications may be made to
adapt a particular situation or material to the teachings of the
invention without departing from the essential scope thereof.
Therefore, it is intended that the invention not be limited to the
particular embodiment(s) disclosed, but that the invention will
include all embodiments falling within the scope of the appended
claims.
* * * * *